Basic invariants
Galois action
Roots of defining polynomial
The roots of $f$ are computed in an extension of $\Q_{ 53 }$ to precision 5.
Minimal polynomial of a generator $a$ of $K$ over $\mathbb{Q}_{ 53 }$: $ x^{2} + 49 x + 2 $
Roots:
| $r_{ 1 }$ |
$=$ |
$ 27 a + 28 + \left(34 a + 7\right)\cdot 53 + \left(38 a + 39\right)\cdot 53^{2} + \left(34 a + 24\right)\cdot 53^{3} + \left(38 a + 33\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 2 }$ |
$=$ |
$ 47 a + 30 + \left(52 a + 39\right)\cdot 53 + \left(29 a + 39\right)\cdot 53^{2} + \left(17 a + 32\right)\cdot 53^{3} + \left(16 a + 17\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 3 }$ |
$=$ |
$ 26 a + 30 + \left(18 a + 12\right)\cdot 53 + 14 a\cdot 53^{2} + \left(18 a + 19\right)\cdot 53^{3} + \left(14 a + 47\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 4 }$ |
$=$ |
$ 6 a + 6 + 45\cdot 53 + 23 a\cdot 53^{2} + \left(35 a + 20\right)\cdot 53^{3} + \left(36 a + 12\right)\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 5 }$ |
$=$ |
$ 5 + 16\cdot 53 + 35\cdot 53^{2} + 31\cdot 53^{3} + 5\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
| $r_{ 6 }$ |
$=$ |
$ 10 + 38\cdot 53 + 43\cdot 53^{2} + 30\cdot 53^{3} + 42\cdot 53^{4} +O\left(53^{ 5 }\right)$ |
Generators of the action on the roots
$r_1, \ldots, r_{ 6 }$
| Cycle notation |
| $(2,4)$ |
| $(1,3)$ |
| $(5,6)$ |
| $(1,2,5)(3,4,6)$ |
Character values on conjugacy classes
| Size | Order | Action on
$r_1, \ldots, r_{ 6 }$
| Character value |
| $1$ | $1$ | $()$ | $3$ |
| $1$ | $2$ | $(1,3)(2,4)(5,6)$ | $-3$ |
| $3$ | $2$ | $(1,3)$ | $1$ |
| $3$ | $2$ | $(1,3)(2,4)$ | $-1$ |
| $4$ | $3$ | $(1,2,5)(3,4,6)$ | $0$ |
| $4$ | $3$ | $(1,5,2)(3,6,4)$ | $0$ |
| $4$ | $6$ | $(1,4,6,3,2,5)$ | $0$ |
| $4$ | $6$ | $(1,5,2,3,6,4)$ | $0$ |
The blue line marks the conjugacy class containing complex conjugation.
